Comment by LegionMammal978
5 days ago
In general, I find that minimax approximation is an underappreciated tool, especially the quite simple Remez algorithm to generate an optimal polynomial approximation [0]. With some modifications, you can adapt it to optimize for either absolute or relative error within an interval, or even come up with rational-function approximations. (Though unfortunately, many presentations of the algorithm use overly-simple forms of sample point selection that can break down on nontrivial input curves, especially if they contain small oscillations.)
They teach a lot of Taylor/Maclaurin series in Math classes (and trig functions are sometimes called "CORDIC" which is an old method too) but these are not used much in actual FPUs and libraries. Maybe we should update the curricula so people know better ways.
Taylor series makes a lot more sense in a math class, right? It is straightforward and (just for example), when you are thinking about whether or not a series converges in the limit, why care about the quality of the approximation after a set number of steps?
Not quite. The point of Taylor’s theorem is that the n-th degree Taylor polynomial around a is the best n-th degree polynomial approximation around a. However, it doesn’t say anything about how good of an approximation it is further away from the point a. In fact, in math, when you use Taylor approximation, you don’t usually care about the infinite Taylor series, only the finite component.
Taylor series have a quite different convergence behavior than a general polynomial approximation. Or polynomial fit for that matter. Many papers were written which confuse this.
For example, 1/(x+2) has a pole at x=-2. The Taylor series around 0 will thus not converge for |x|>2. A polynomial approximation for, say, a range 0<x<L, will for all L.
Not sure I would call Remez "simple"... it's all relative; I prefer Chebyshev approximation which is simpler than Remez.
Perhaps, but at least I find it very simple for the optimality properties it gives: there is no inherent need to say, "I know that better parameters likely exist, but the algorithm to find them would be hopelessly expensive," as is the case in many forms of minimax optimization.
Ideally either one is just a library call to generate the coefficients. Remez can get into trouble near the endpoints of the interval for asin and require a little bit of manual intervention, however.